Аттестация и самооценка начались сегодня.

Because the subject is actually two things together: аттестация и самооценка.

In Russian, when two singular nouns are joined by и and act as the subject, the verb normally goes in the plural:

• Аттестация и самооценка начались сегодня.
  The assessment and the self-evaluation started today.

If there were only one subject, you would use singular:

• Аттестация началась сегодня.
• Самооценка началась сегодня.

So начались agrees with the whole compound subject, not just with the first noun.

Because начались comes from the verb начаться, which means to begin / to start.

Russian often uses:

• начать / начинать = to start something
• начаться / начинаться = to start, to begin (for the event itself)

Compare:

• Они начали аттестацию.
  They started the assessment.
  Here someone is doing the action to something else.

• Аттестация началась.
  The assessment started.
  Here the event itself began.

So in your sentence, начались is the correct form because аттестация и самооценка are the things that began.

The -ся ending marks a reflexive verb, but in many Russian verbs it does not literally mean oneself.

Here, начаться is just the normal intransitive counterpart of начать:

• начать = to start something
• начаться = to begin, to start

So начались does not mean started themselves in a literal English sense. It just means began or started.

This is very common in Russian:

• открыть = to open something
• открыться = to open
• изменить = to change something
• измениться = to change

They are in the nominative case, because they are the subject of the sentence.

The sentence is telling you what started, and the things that started are:

• аттестация
• самооценка

Both are singular feminine nouns in their dictionary form:

• аттестация
• самооценка

Even though each noun is singular, together they form a plural subject, which is why the verb is plural: начались.

Yes.

• аттестация is feminine
• самооценка is feminine

You can tell from their endings:

• nouns ending in -а / -я are very often feminine

That is why, if each one appeared alone, the past-tense verb would be feminine singular:

• Аттестация началась.
• Самооценка началась.

But together they take plural:

• Аттестация и самооценка начались.

Not always. Аттестация can mean different kinds of formal evaluation depending on context, such as:

• assessment
• evaluation
• attestation
• certification
• performance review

In workplaces or schools, it often refers to some kind of official review or evaluation process.

So a learner may need context to choose the best English translation. Even if the meaning is already given, it is useful to know that аттестация is a fairly formal word.

Not necessarily.

Самооценка can mean:

• self-assessment
• self-evaluation
• self-esteem

The exact meaning depends on context.

In this sentence, because it is paired with аттестация, it most likely means self-assessment or self-evaluation, not emotional self-esteem.

So the sentence probably refers to two evaluation processes that began today.

Yes, Russian word order is flexible. Сегодня can appear in different places:

• Аттестация и самооценка начались сегодня.
• Сегодня аттестация и самооценка начались.
• Аттестация и самооценка сегодня начались.

All of these are possible, though they may sound slightly different in emphasis.

The version with сегодня at the end is neutral and natural. It often places a little focus on when the action happened.

Here it functions as an adverb of time, meaning today.

It answers the question when?

• Когда начались аттестация и самооценка?
• Сегодня.

So in this sentence, you can think of сегодня simply as today.

Начались is:

• past tense
• plural
• from a perfective verb: начаться

Because it is perfective, it presents the action as a completed event boundary: they began / they started.

If you wanted an imperfective idea such as were beginning or a more ongoing repeated sense, Russian would use начинаться instead in the appropriate form.

So начались сегодня means something like started today as a complete event.

The stress is:

• аттеста́ция
• самооце́нка
• начали́сь
• сего́дня

Stress matters in Russian, so it is worth learning words with their stress from the beginning.

A rough pronunciation guide:

• аттеста́ция = aht-teh-STA-tsi-ya
• самооце́нка = sa-ma-a-TSEN-ka
• начали́сь = na-cha-LEES
• сего́дня = si-VOD-nya

Sometimes, yes, but it depends on context.

If the context already makes it clear what is being discussed, Russian could say:

• Они начались сегодня.
  They started today.

But without context, that would be vague. The full sentence names both subjects clearly:

• Аттестация и самооценка начались сегодня.

Russian often omits things only when they are already obvious from the conversation.